Phenomenology of modified gravity at recombination

We discuss the phenomenological imprints of modifications to gravity in the early Universe with a specific focus on the time of recombination. We derive several interesting results regarding the effect that such modifications have on cosmological observables, especially on the driving and phasing of acoustic oscillations, observed in the cosmic microwave background (CMB) and baryon acoustic oscillations, as well as the weak gravitational lensing of the CMB and of galaxy shapes. This widens the pool of measurements that can be used to test gravity with present and future surveys, in particular realizing the full constraining power of the structure of the primary peaks of the CMB spectrum. We investigate whether such a phenomenology can relax tensions between cosmological measurements and find that a modification of the gravitational constant at recombination would help in reconciling measurements of the CMB with local measurements of the Hubble constant.

Figure 1

The comparison of the Weyl potential evolution between our MG example models and GR. The four panels represent models with ${\mu }_{\infty }=1.2$, ${\gamma }_{\infty }=1.2$, ${\mu }_0=1.2$ and ${\gamma }_0=1.2$, respectively. The three vertical dashed lines indicate, from left to right, respectively, matter-radiation equality, the transition of the MG parameters (here $z=30$, see definition in Sec. 3a), and $\mathrm{\Lambda }$-matter equality. Before horizon crossing, $\mu$ has a limited effect on the evolution of Weyl potential due to the small fraction of neutrino energy density, while a larger $\gamma$ decreases its amplitude in both radiation and matter dominated epochs and slows the potential decay in the acceleration epoch. When crossing the horizon during radiation epoch, a larger $\mu$ delays the potential decay significantly while the same change in $\gamma$ leads to a small effect. After horizon crossing, a larger $\mu$ increases the amplitude of the potential due to a larger effective gravitational constant $\mu G$, while $\gamma$ does not affect the subhorizon evolution. For details of the early and late time behaviors and the effect of the transition, see the discussion in Sec. 3c.

Figure 2

The fractional change in transfer functions relative to their GR values at redshift $z=0$ due to MG of (a) the Weyl potential and (b) the synchronous gauge matter density perturbations. Different colors represent different example models as in Fig. 1, as shown in legend. The vertical lines show the scales, $(k_{\mathrm{DE}},k_{\mathrm{T}},k_{\mathrm{eq}})$, corresponding to the modes crossing the horizon at $\mathrm{\Lambda }$-matter equality, transition in the MG functions, and matter-radiation equality, respectively.

Figure 3

The CMB anisotropy source functions in $k$-space in units of amplitude of primordial comoving curvature perturbation in two MG example models with ${\mu }_{\infty }=1.2$ and ${\gamma }_{\infty }=1.2$ and GR. Different lines correspond to different physical effects and models, as shown in figure and legend. The vertical dashed line shows a mode that crosses the horizon at recombination ($z_{*}$).

Figure 4

The fractional change in the unlensed CMB temperature spectrum in two MG example models with ${\mu }_{\infty }=1.2$ and ${\gamma }_{\infty }=1.2$ relative to the GR spectrum. The vertical solid lines indicate the angular position of the GR peaks of the unlensed CMB spectrum. Notice that the variations are mainly out of phase with the peaks.

Figure 5

The CMB lensing potential power spectrum in the harmonic domain. Different colors correspond to different models as shown in legend.

Figure 6

The fractional change in ${\sigma }_8^{\mathrm{WL}}$ from its the GR value. Different colors correspond to different models as shown in legend.

Figure 7

The CDM spatial linear correlation function $\xi (r)$. Different colors correspond to different models as shown in legend. Vertical lines represent the BAO peak which are slightly shifted by early time MG parameters.

Figure 8

The marginalized joint posterior of the parameters of the late time MG model, ${\mu }_0+{\gamma }_0$. $\mathrm{\Lambda }\mathrm{CDM}$ results are also added for comparison. Different colors correspond to different combination of datasets, as shown in legend. The darker and lighter shades correspond, respectively, to the 68% C.L. and the 95% C.L. Dashed lines indicate the GR limit of the MG parameters.

Figure 9

The Planck TT residuals for the best fit late time modified gravity model, relative to the best fit $\mathrm{\Lambda }\mathrm{CDM}$ CMBTT model. The foreground parameters are fixed to the $\mathrm{\Lambda }\mathrm{CDM}$ best fit values for compatibility with the foreground model removed from the data. Different colors correspond to different combination of datasets, as shown in legend. The residuals are normalized to ${\sigma }_{\mathrm{CV}}$, the cosmic variance per multipole of the best fit $\mathrm{\Lambda }\mathrm{CDM}$ CMBTT model (see text). The vertical solid lines indicate the angular position of acoustic peaks of the unlensed spectrum in the $\mathrm{\Lambda }\mathrm{CDM}$ model.

Figure 10

The WL-only and CMBlens-only constraints on ${\mu }_0$ and ${\gamma }_0$ while keeping all other parameters fixed to their best fit $\mathrm{\Lambda }\mathrm{CDM}$ values for CMBTT. The continuous lines show the iso-contours of ${\sigma }_8^{\mathrm{WL}}{\mathrm{\Omega }}_m^{0.5}$. The darker and lighter shades correspond, respectively, to the 68% C.L. and the 95% C.L. Dashed lines indicate the GR limit of the MG parameters.

Figure 11

The CMBTT constraints on the cosmological parameters in the ${\mu }_{\infty }$ only model. Results of $\mathrm{\Lambda }\mathrm{CDM}$ model are also added for comparison. The darker and lighter shades correspond, respectively, to the 68% C.L. and the 95% C.L. Dashed lines indicate the GR limit of the MG parameters.

Figure 12

TT residuals relative to the best fit $\mathrm{\Lambda }\mathrm{CDM}$ CMBTT model, similar to Fig. 9 but with the best fit ${\mu }_{\infty }$-only model to $\mathrm{CMBTT}+\mathrm{H}0+\mathrm{WL}$ datasets with $H_0\approx 70$ (red line). A model with the same cosmological parameters but ${\mu }_{\infty }=1$ (orange line) shows that ${\mu }_{\infty }$ compensates for the oscillatory fluctuations of the high $H_0$ model that are disfavored by the data.

Figure 13

The marginalized joint posterior for the parameters in the ${\mu }_{\infty }$ only model. Different colors correspond to different combination of datasets, as shown in legend. The darker and lighter shades correspond, respectively, to the 68% C.L. and the 95% C.L. The dashed lines indicate the value of ${\mu }_{\infty }$ in GR limit.

Figure 14

CMB lens power spectrum residuals relative of the ${\mu }_{\infty }$ only model to the best fit $\mathrm{\Lambda }\mathrm{CDM}$ CMBTT model. The models are the same as in Fig. 12 as also shown in the legend. Again ${\mu }_{\infty }$ compensates the changes due to the larger $H_0$ value.

Figure 15

TE residuals relative to the best fit $\mathrm{\Lambda }\mathrm{CDM}$ CMBTT model, similar to Fig. 12 but with TE spectrum. Again ${\mu }_{\infty }$ compensates the changes due to the larger $H_0$ value especially around the first minimum where the data also fluctuate low (leftmost vertical line).

Figure 16

The marginalized joint posterior for ${\mu }_{\infty }$ and $H_0$ in the ${\mu }_{\infty }$ only model. Different colors correspond to different combination of datasets, as shown in legend. The darker and lighter shades correspond, respectively, to the 68% C.L. and the 95% C.L. The dashed lines indicate the value ${\mu }_{\infty }$ in GR limit.

Figure 17

The marginalized joint posterior for ${\gamma }_{\infty }$ and $H_0$ in the ${\gamma }_{\infty }$ only model with the CMBTT dataset. The darker and lighter shades correspond, respectively, to the 68% C.L. and the 95% C.L. The dashed lines indicate the value ${\gamma }_{\infty }$ in GR limit where nearly the same range in $H_0$ is allowed.

Figure 18

The marginalized joint posterior for the parameters in the ${\mu }_{\infty }+{\mu }_0$ only model. Different colors correspond to different combination of datasets, as shown in legend. The darker and lighter shades correspond, respectively, to the 68% C.L. and the 95% C.L. The dashed lines indicate the values of the MG parameters in the GR limit.

Figure 19

The marginalized joint posterior for ${\gamma }_{\infty }$ and ${\gamma }_0$ in the ${\gamma }_{\infty }+{\gamma }_0$ model with the CMBTT dataset. The darker and lighter shades correspond, respectively, to the 68% C.L. and the 95% C.L. The dashed lines indicate the values of the MG parameters in the GR limit.

Figure 20

The comparison of the unlensed large scale CMB temperature spectrum in several MG example models with different transition widths relative to the GR spectrum. Left: early time MG example models (${\mu }_{\infty }=1.2$). Right: late time MG example models along the degeneracy direction (${\mu }_0=0.9$, ${\gamma }_0=1.7$). Here we also show the erroneously large effect predicted from the inconsistent implementation of an instantaneous transition employed in MGCAMB.

Figure 21

The CMBTT constraints on late time MG parameters with different transition treatments from GR to MG. The darker and lighter shades correspond, respectively, to the 68% C.L. and the 95% C.L. The dashed lines indicate the values of MG parameters in GR limit. With a sharp transition the ISW effect shown in Fig. 20 breaks the degeneracy between parameters leading to unrealistically strong constraints. The inconsistent instantaneous transition gives even stronger, but incorrect, constraints.